Download Rational points on the Cantor middle thirds set

RATIONAL POINTS IN THE CANTOR MIDDLE THIRDS SET
RHIANNON SCHAYER, DAVID JORDAN
Abstract. In this paper we develop methods for determining the asymptotic
behavior of rational points on the Cantor middle thirds set. Particularly we
are concerned with how many reduced rational points with denominators from
1 to N lie on the set.
1. Definitions
To facilitate our exploration of rational points in the Cantor middle thirds set, we
have included several key definitions.
Definition 1.1. A rational number pq ∈ Q is a Cantor rational if it meets the
following conditions:
(1) pq is in the Cantor middle thirds set, as defined in [1]
(2) p and q are relatively prime, i.e. pq is in reduced form.
Denote the Cantor rationals by the symbol MQ . Hereafter, abbreviate “ qp is a
Cantor rational” by pq ∈ MQ .
Because we are working towards estimating the total number of Cantor rationals
with denominator q, where 1 ≤ q ≤ N , we must introduce the notation we will use
to denote this sum.
Definition 1.2. Define An to be the set of Cantor rationals with denominator n.
Define |An | to be the number of elements in An .
Definition 1.3. Define Cn to be the total number of Cantor rationals with denominator q, where 1 ≤ q ≤ N .
Cn =
N
X
i=1
|Ai |
The Cantor middle thirds set is constructed in a way that favors the use of base
three over base ten. Therefore, we introduce the following definition.
Definition 1.4 ([1]). The sequence of integers 0.s1 s2 s3 . . . where each si is either
0, 1, or 2 is called the ternary expansion of x if
∞
X
si
x=
i
3
i=1
Representing numbers in this way provides us with a simple test for Cantor rationality. If a number’s ternary expansion is eventually periodic, then that number is
rational. If, further, that number has a ternary expansion containing no ones, then
it is a Cantor rational. See [1] for more details.
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RHIANNON SCHAYER, DAVID JORDAN
Note that some values on [0, 1] have multiple ternary expansions. In cases where
the period is 0 or 2 there is a potential ambiguity. For example,
1 = 1.000 . . .
= 0.222 . . .
In such cases, we choose the expansion containing no ones, if it exists. If both
possible expansions of a number contain ones, then that number is not a Cantor
rational. In the above example we may choose the second expansion, which contains no ones, and thus 1 ∈ MQ .
In addition to using the ternary expansion of a number, we also found the 3-adic
expansion to be a useful tool for determining Cantor rationality. It is important
to note that using p-adic numbers was not necessary to our theorems (the reader
may wish to develop the proofs for themselves using simpler considerations), but
using p-adics made the proofs clearer and shorter. We will now briefly introduce
the reader to the notion of p-adic numbers as described in [3].
Definition 1.5 ([3]). Let p ∈ N be any prime number. Define a map | · |p on Q as
follows:
1
if x 6= 0
pordp x
|x|p =
0
if x = 0,
where
the highest power of p which divides x, if x ∈ Z
ordp x =
ordp a − ordp b,
if x = ab , a, b ∈ Z, b 6= 0.
Call this map the p-adic norm.
Define Qp to be the completion of Q with respect to the p-adic norm. We call
Qp the field of p-adic numbers, where the elements of this field are equivalence
classes of Cauchy sequences in Q with respect to the extension of the p-adic norm.
Specifically, we will work with 3-adic numbers.
Definition 1.6. The sequence of integers d−m . . . d−1 .d0 d1 d2 d3 . . ., where each di
is either 0,1, or 2 is called the 3-adic expansion of a if
∞
X
a=
di pi .
i=−m
We know that the 3-adic expansion represents a rational number if and only if it is
eventually periodic to the right [3].
There is no ambiguity in 3-adic expansions like that which we encountered with
ternary expansions. It is a property of the p-adic metric that if two p-adic series
converge to the same number, then they are the same series. Since the 3-adic
expansion is just one such series, the 3-adic expansion of a number is unique [3].
We will frequently call on the fact that the ord3 of a number is equivalent to the
number of leading zeros in its 3-adic expansion.
In the terminology of the p-adic numbers, an expansion with no digits to the left
3
of the decimal represents a 3-adic integer. Many numbers which are not integers in
R are integers in Qp . For instance,
1
= .1220
13
is an integer in the 3-adic field.
2. The 3-adic field and the Cantor rationals
With the following theorem, we intend to show that the test for Cantor rationality that we applied to ternary expansions can also be similarly applied to 3-adic
expansions.
Theorem 2.1. Let q be not divisible by three and p ≤ q. Then pq is a Cantor
rational if and only if the 3-adic expansion of − pq contains no ones.
Proof. Since − pq is rational, with q not divisible by three, it has a purely periodic
ternary expansion, − qp = −0.a1 a2 . . . an , for some n. From elementary number
theory, this means we may write
a1 3n−1 + a2 3n−2 + · · · + an
3n − 1
1
n−2
n−1
a
+
·
·
·
+
a
3
+
a
3
.
=
n
2
1
1 − 3n
In the field of 3-adic numbers, we have that
∞
X
1
=
3ni .
1 − 3n
i=0
−
p
q
= −
Note that this sum is only convergent for 3-adic numbers. Thus,
!
∞
X
p
ni
−
=
3
an + · · · + a2 3n−2 + a1 3n−1
q
i=0
= .an . . . a1 ,
when expanded 3-adically. From this we see that the 3-adic expansion of − pq contains a one if and only if the ternary expansion of pq does, and we are done.
3. Some Helpful Observations
Lemma 3.1. The following facts hold for rational points on the Cantor middle
thirds set:
(1) The Cantor middle thirds set is invariant under reflection around the point
p
q−p
p
1
2 . We call the reflection, R( q ) = q , the complement of q .
(2) The Cantor middle thirds set is invariant under division and multiplication
by three, so long as the product still lies in the interval [0, 1].
Proof. (1) follows immediately from the symmetry of the set. (2) follows from the
fact that multiplication and division by three simply translates the decimal point
to the right or left, respectively, and thus preserves the no-ones condition. So long
as multiplying by three does not land us outside our interval, it preserves our condition for Cantor rationality.
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RHIANNON SCHAYER, DAVID JORDAN
Corollary 3.2. Aq is nonempty if and only if A3q is nonempty.
4. Spawning Sets
We have described several general properties of Cantor rationals and have discussed
the two ways that we will be representing them - 3-adically and in ternary. We
continue with a definition that follows from the facts presented thus far, specifically
from Corollary 3.2.
Definition 4.1. Define Aq to be a spawning set for the Cantor rationals if Aq is
non-empty, and q is not divisible by 3. Define the sets, A3q , A9q , A27q , A81q , . . . to
be the child sets of Aq .
After proving the relatively simple observations above, we ran several computer
experiments and noticed that the elements in the spawning sets had a particular
structure: exactly half of the elements had numerators divisible by three and these
elements corresponded by reflection to those elements with numerators not divisible
by three. The following theorem establishes that exactly half the elements of a
spawning set are divisible by three.
Theorem 4.2. Let Aq be a spawning set. Then pq ∈ Aq has p divisible by three if
and only if its complement, q−p
q , has (q-p) not divisible by three.
Proof. (⇒) Let p be divisible by three. Then using simple number theory, we know
that if 3 6 | q then 3 6 | (q − p).
(⇐) Conversely, let p be not divisible by three. Then certainly -p is also not
divisible by three, and so it will have a non-zero number for its first 3-adic digit.
Thus,
p
ord3 (− ) = ord3 (−p) − ord3 (q) = ord3 (−p) = ord3 (p) = 0,
q
which implies that pq will also have a non-zero first 3-adic digit. Theorem 2.1
dictates, then, that this digit be a two. Adding one to both sides, it is clear that
R( pq ) = q−p
q has a zero for its first 3-adic digit, and thus,
ord3 (q − p) = ord3 (q − p) − ord3 (q) = ord3 (
q−p
) > 0,
q
as required.
Note that if Aq is a child set, then it contains no
p
q with p divisible by three, because it must have q divisible by three, and since all
p
q ∈ Aq must be in reduced form, p cannot be divisible by three.
We have thus far discussed some basic facts about rational points in the Cantor set and about rationals in the spawning sets. At this point,we are concerned
with two questions: for which q is Aq a spawning set, and how many elements does
each spawning set contain? We will start our exploration by examining for which
general q Aq is a spawning set.
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Theorem 4.3. A 3n −1 is a spawning set for all integers n, and all φ ∈ Z such that:
φ
φ | 3n − 1
n−1
X
φ|
ai 3i for some combination of ai = 0 or 2.
(1)
(2)
i=0
P
(3)
n−1
i=0
ai 3i /φ
(3n − 1)/φ
is reduced.
Furthermore, all spawning sets are of this form for some φ.
n
Proof. Clearly 3 φ−1 is not divisible by three, and so it suffices to show that A 3n −1
φ
is nonempty. To show this, we need only consider the definition of the ternary
expansion. If a pq has period n, this means that we may write
p
q
=
Pn−1
i=0 ai 3
3n − 1
i
.
where our condition for Cantor rationals is that each ai be either a zero or a two.
Since we assume that φ divides both the numerator and the denominator, for some
combination of ai , and we assume that after dividing both by φ, we have a reduced
rational, we have that A 3n −1 is nonempty, and thus it is a spawning set for all
φ
integers n and φ satisfying (1)-(3).
To show that all spawning sets are of this form, assume pq is an element of a spawning set. Then it has a purely periodic ternary expansion, and thus from elementary
number theory we may write
Pn−1
ai 3i
p
= i=0
,
n
q
3 −1
where the ai are the digits in the period of pq .
Though we cannot identify all φ that will yield a spawning set in the above formula,
we can definitively say that the following two formulas will be spawning sets for all
integers n.
Corollary 4.4. A 3n −1 is a spawning set for all integers n.
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Proof. Note that 3n − 1 is always even, as are all the ai . If we take the specific case,
a0 = 2, ai = 0 ∀ i 6= 0, φ = 2, we get that 3n1−1 ∈ Mq . Thus A 3n −1 is a spawning
2
set for all n.
Theorem 4.5. A3n +1 is a spawning set for all integers n.
Proof. Certainly 3n + 1 is not divisible by three, and so it suffices to show that
A3n +1 contains at least one Cantor rational. We divert our attention momentarily
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RHIANNON SCHAYER, DAVID JORDAN
to ternary expansions where the period is of the following form: Let a˘i = 2 − ai .
0.a1 a˘1
0.a1 a2 a˘1 a˘2
0.a1 a2 a3 a˘1 a˘2 a˘3
..
.
0.a1 . . . an a˘1 . . . a˘n
If we restrict each ai = 0 or 2, then certainly sequences of this form represent Cantor
rationals. We will see that these are precisely the expansions for 3np+1 . Recalling
the definition of the rational period, we may write the general case as follows:
3n (a1 . . . an ) + 3n − 1 − a1 . . . an
32n − 1
n
(3 − 1) (a1 . . . an ) + (3n − 1)
=
(3n + 1)(3n − 1)
a1 . . . an + 1
.
=
3n + 1
However, we cannot be sure that all combinations for the numerator will yield a
rational in reduced form (in most cases, they won’t). But since ai = 0 ∀ i gives
1
n
n
3n +1 , we know that A3 +1 has at least one element, and thus A3 +1 is a spawning
set for all n.
0.a1 . . . an a˘1 . . . a˘n
=
Unfortunately we cannot answer in closed form the question of divisibility, and
so we will only be able to predict the occurrence of spawning sets approximately.
For the time being, then, let us turn our attention to the question of how many
elements a given spawning set will contain.
5. Results
Building on what we have established so far about spawning sets and their elements,
we are ready to make a statement about how the number of Cantor rationals with
denominators less than a given value will grow. Specifically we develop a strict
minimum, which we can use to describe the asymptotic behavior of the Cantor
rationals.
Theorem 5.1. Let Aq be a spawning set. Then |A3k q | = 2k−1 |Aq |. for k ≥ 1.
Proof. Let Aq be a spawning set.
Case 1: Let k = 1. By Theorem 4.2, we know that precisely half the elements
of Aq are divisible by three, and half are not. Thus, when we apply Theorem ??
to those that are divisible by three, they will correspond to other elements in Aq .
And when we apply Theorem ?? to those elements of Aq that are not divisible
by three, they will each correspond to a unique element in the child set, A3q . By
Theorem ??, each of these elements has a unique complement, also in A3q . Thus,
|A3q | = 2 · 12 · |Aq | = |Aq |.
Case 2: Let k > 1. By lemma ?? we know that none of the Cantor rationals in the
child sets can be divisible by three. So, unlike in the case of spawning sets, Theorem
7
?? gives that all of the elements in A3k q yield a unique element in A3k+1 q . Again,
by Theorem ??, each of these new elements has a unique complement in A3k+1 q . So
each element in A3k q corresponds to two elements in A3k+1 q by this construction,
and so |A3k+1 q | = 2|A3k q |. Combining cases 1 and 2, we have |A3k q | = 2k−1 |Aq |. From these two theorems, we can calculate precisely a very useful quantity. Let A i
be a spawning set, and take ΩN
i to be the sum,
ΩN
i = |Ai | + |A3i | + |A9i | + . . . + |A
3blog3 (
N )c
i
i
|.
This sums the contributions of the spawning set Ai and all its child sets with
denominators less than N . We can simplify ΩN
i rather nicely.
ΩN
i
= |Ai | + |A3i | + |A9i | + . . . + |A
= |Ai | + |Ai | + 2|Ai | + . . . + 2


blog3 ( N
i )c−1
X
= |Ai | 1 +
2k 
N
3blog3 ( i )c i
blog3 ( N
i )c−1
|
|Ai |
k=0
= |Ai |2
blog3 ( N
i )c
If we observe that bxc ≥ x − 1, we get a very nice formula for ΩN
i .
Ωni
N
= |Ai |2blog3 ( i )c
N
≥ |Ai |2log3 ( i )−1
|Ai |
N log 2
=
exp(log( )
)
2
i log 3
log 2
log 2
1 log
=
N 3 |Ai |i− log 3 .
2
Then, our original quantity, CN is simply the sum of all the Ω’s,
CN ≥
N
log 2
2 X
1 log
|Si |i− log 3 ,
N log 3
2
i=1
where here the Si are spawning sets. The next task will be to discover the asymptotic behavior of this quantity.
References
[1] Devaney, Robert L. A First Course in Chaotic Dynamical Systems Addison Wesley
Publishing Company, Ltd., 1992.
[2] Rosen, Kenneth H. Elementary Number Theory and its applications, Addison Wesley
Longman, Inc., 2000.
[3] Katok, Svetlana. p-adic analysis in comparison with real, MASS Selecta: Teaching and
Learning Advanced Undergraduate Mathematics, American Mathematical Society,
2003.